Comparison of Wide and Compact Fourth Order Formulations of the Navier-Stokes Equations

In this study the numerical performances of wide and compact fourth order formulation of the steady 2-D incompressible Navier-Stokes equations will be investigated and compared with each other. The benchmark driven cavity flow problem will be solved using both wide and compact fourth order formulations and the numerical performances of both formulations will be presented and also the advantages and disadvantages of both formulations will be discussed

Numerical Solutions of 2-D Steady Incompressible Driven Cavity Flow at High Reynolds Numbers

Numerical calculations of the 2-D steady incompressible driven cavity flow are presented. The Navier-Stokes equations in streamfunction and vorticity formulation are solved numerically using a fine uniform grid mesh of 601x601. The steady driven cavity solutions are computed for Re<21,000 with a maximum absolute residuals of the governing equations that were less than 10-10. A new quaternary vortex at the bottom left corner and a new tertiary vortex at the top left corner of the cavity are observed in the flow field as the Reynolds number increases. Detailed results are presented and comparisons are made with benchmark solutions found in the literature

Fine Grid Numerical Solutions of Triangular Cavity Flow

Numerical solutions of 2-D steady incompressible flow inside a triangular cavity are presented. For the purpose of comparing our results with several different triangular cavity studies with different triangle geometries, a general triangle mapped onto a computational domain is considered. The Navier-Stokes equations in general curvilinear coordinates in streamfunction and vorticity formulation are numerically solved. Using a very fine grid mesh, the triangular cavity flow is solved for high Reynolds numbers. The results are compared with the numerical solutions found in the literature and also with analytical solutions as well. Detailed results are presented

Discussions on Driven Cavity Flow

The widely studied benchmark problem, 2-D driven cavity flow problem is discussed in details in terms of physical and mathematical and also numerical aspects. A very brief literature survey on studies on the driven cavity flow is given. Based on the several numerical and experimental studies, the fact of the matter is, above moderate Reynolds numbers physically the flow in a driven cavity is not two-dimensional. However there exist numerical solutions for 2-D driven cavity flow at high Reynolds numbers

Numerical Solutions of 2-D Steady Incompressible Flow in a Driven Skewed Cavity

The benchmark test case for non-orthogonal grid mesh, the "driven skewed cavity flow", first introduced by Demirdzic et al. (1992, IJNMF, 15, 329) for skew angles of alpha=30 and alpha=45, is reintroduced with a more variety of skew angles. The benchmark problem has non-orthogonal, skewed grid mesh with skew angle (alpha). The governing 2-D steady incompressible Navier-Stokes equations in general curvilinear coordinates are solved for the solution of driven skewed cavity flow with non-orthogonal grid mesh using a numerical method which is efficient and stable even at extreme skew angles. Highly accurate numerical solutions of the driven skewed cavity flow, solved using a fine grid (512x512) mesh, are presented for Reynolds number of 100 and 1000 for skew angles ranging between 15<alpha<165

Numerical Performance of Compact Fourth Order Formulation of the Navier-Stokes Equations

In this study the numerical performance of the fourth order compact formulation of the steady 2-D incompressible Navier-Stokes equations introduced by Erturk et al. (Int. J. Numer. Methods Fluids, 50, 421-436) will be presented. The benchmark driven cavity flow problem will be solved using the introduced compact fourth order formulation of the Navier-Stokes equations with two different line iterative semi-implicit methods for both second and fourth order spatial accuracy. The extra CPU work needed for increasing the spatial accuracy from second order (O(x2)) to fourth order (O(x4)) formulation will be presented

Link-wise Artificial Compressibility Method

The Artificial Compressibility Method (ACM) for the incompressible Navier-Stokes equations is (link-wise) reformulated (referred to as LW-ACM) by a finite set of discrete directions (links) on a regular Cartesian mesh, in analogy with the Lattice Boltzmann Method (LBM). The main advantage is the possibility of exploiting well established technologies originally developed for LBM and classical computational fluid dynamics, with special emphasis on finite differences (at least in the present paper), at the cost of minor changes. For instance, wall boundaries not aligned with the background Cartesian mesh can be taken into account by tracing the intersections of each link with the wall (analogously to LBM technology). LW-ACM requires no high-order moments beyond hydrodynamics (often referred to as ghost moments) and no kinetic expansion. Like finite difference schemes, only standard Taylor expansion is needed for analyzing consistency. Preliminary efforts towards optimal implementations have shown that LW-ACM is capable of similar computational speed as optimized (BGK-) LBM. In addition, the memory demand is significantly smaller than (BGK-) LBM. Importantly, with an efficient implementation, this algorithm may be one of the few which is compute-bound and not memory-bound. Two- and three-dimensional benchmarks are investigated, and an extensive comparative study between the present approach and state of the art methods from the literature is carried out. Numerical evidences suggest that LW-ACM represents an excellent alternative in terms of simplicity, stability and accuracy.Comment: 62 pages, 20 figure